In this paper, using the exact expression for the pairwise error probability derived in terms of the message symbol distance between two message vectors rather than the codeword symbol distance between two transmitted codeword matrices, the exact closed form expressions for the symbol error probability of any linear orthogonal space-time block codes in slow Rayleigh fading channel are derived for QPSK, 16-QAM, 64-QAM, and 2 56-QAM.

A family F of min-wise independent permutations is known to be a useful tool of indexing replicated documents on the Web. We say that a family F of permutations on {0,1,. . . ,n-1} is min-wise independent if for any X {0,1,. . . ,n-1} and any x X, Pr[min {π(X)} = π(x)]= ||X||-1 when π is chosen uniformly at random from F, where ||A|| is the cardinality of a finite set A. We also say that a family F of permutations on {0,1,. . . ,n-1} is d-wise independent if for any distinct x1,x2,. . . ,xd {0,1,. . . , n-1} and any distinct y1,y2,. . . ,yd {0,1,. . . , n-1}, Pr[i=1d π(xi) = π(yi)]= 1/{n(n-1) (n-d+1)} when π is chosen uniformly at random from F (note that nontrivial constructions of d-wise independent family F of permutations on {0,1,. . . ,n-1} are known only for d=2,3). Recently, Broder, et al. showed that any family F of pairwise (2-wise) independent permutations behaves close to a family of min-wise independent permutations, i.e., for any X {0,1,. . . ,n-1} such that 3 ||X||=k n-2 and any x X, (lower bound) Pr[min {π(X)}=π(x)] 1/{2(k-1)}; (upper bound) Pr[min {π(X)}=π(x)] O(1/k). In this paper, we extend these bounds to 3-wise independent permutation family and show that any family of 3-wise independent permutations behaves closer to a family of min-wise independent permutations, i.e., for any X {0,1,. . . ,n-1} such that 4 ||X||=k n-3 and any x X, (lower bound) Pr[min {π(X)}=π(x)] 1/{2(k-2)}- 1/{6(k-2)2}; (upper bound) Pr[min {π(X)}=π(x)] 2/k - 2/k + 1/(3kk).

Chen et al., have proposed a new estimation method for the membership values in fuzzy sets. The proposed scheme takes input from empirical/experimental data, which reflect the expert's knowledge on the relative degree of belonging of the members, and then searches for the best fit membership values of the element. Through the estimation of the practical case (Sect. 3 in [1]) the algorithm suggests to normalize the estimated membership values if there is any among them more than one and change some condition to guarantee its positiveness. In this paper, we show how to use the same imposed condition to guarantee that the estimated membership values will be within the unit interval without normalization.

An efficient algorithm is given for finding all solutions of piecewise-linear resistive circuits containing nonseparable transistor models such as the Gummel-Poon model or the Shichman-Hodges model. The proposed algorithm is simple and can be easily programmed using recursive functions.

Finding DC solutions of nonlinear networks is one of the most difficult tasks in circuit simulation, and many circuit designers experience difficulties in finding DC solutions using Newton's method. Piecewise-linear analysis has been studied to overcome this difficulty. However, efficient piecewiselinear algorithms have not been proposed for nonlinear resistive networks containing the Gummel-Poon models or the Shichman-Hodges models. In this paper, a new piecewise-linear algorithm is presented for solving nonlinear resistive networks containing these sophisticated transistor models. The basic idea of the algorithm is to exploit the special structure of the nonlinear network equations, namely, the pairwise-separability. The proposed algorithm is globally convergent and much more efficient than the conventional simplical-type piecewise-linear algorithms.